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How do we determine if a statement can't be proved by mathematics alone? It seems mathematics can only prove something that can be defined purely in mathematics terms, but can't prove simple statements that can't be defined or expressed in purely mathematics terms, but are there principles or statements that help us determine if a phrase expressed in natural languages can be proven mathematically? A sort of theorems or guiding principles would be nice.

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  • "but can't prove simple statements that can't be defined or expressed in purely mathematics terms" sounds trivial but can be tricky... From one side, if a statement is not in the context of the language of a mathematical theory, why can we imagine that that theory can prove it? Euclidean geometry does not prove facts about e.g. probability. But, in a more general setting, how can we extend the concept of proof outside mathematics? Can we prove "theorems" about history, society, etc? Sep 16, 2022 at 15:08
  • As the great American philosopher Frankie Lymon asked in 1956, Why do fools fall in love? youtube.com/watch?v=2sAHiR0rkJg
    – user4894
    Sep 16, 2022 at 17:33
  • Proving isolated statements is either trivial or nonsensical, it is proving large clusters of statements from few simpler ones (premises) that is of interest. And if you are asking which informal arguments (not statements) are mathematically formalizable into proofs the answer is those that can be algorithmically checked (after including context and disambiguating natural language conventions), see Azzouni, Why Do Informal Proofs Conform to Formal Norms?
    – Conifold
    Sep 17, 2022 at 3:26

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How do we determine if a statement can't be proved by mathematics alone?

No statement is ever proved by mathematics alone. Mathematical proofs need to be logical, and logic is not magic, and all that you can do using logic is to prove not that some statement is true, but that the truth of a statement (the conclusion) follows logically from the truth of another statement (the premises), and then your logical proof does not prove the premises true. So, the premises may be false and so your conclusion may be false, too. Thus, a proof does not prove that the conclusion is true. No logical proof does. If mathematicians claim otherwise, they are just wrong.

One limitation in the applicability of mathematics is the difficulty to understand what the concepts we use in everyday verbal communication well enough to be able to articulate a definition rigorous enough to reason logically about it.

Human beings are obviously able to reason logically but only to a very limited extent, so another obvious limitation is the complexity of the logical reasoning necessary to prove statements.

One good example of that is logic itself. As of today, after 2,500 years of Aristotelian logic and 170 years of mathematical logic, no mathematician has been able to articulate any formal theory correctly modelling human deductive logic.

That being said, it seems that there is no impossibility, as long as one really understands the various concepts involved, but logical proofs are not enough in themselves to provide the necessary understanding. You need something like intuition as well, and this requires going through the data available, i.e., hard work.

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